{
 "cells": [
  {
   "cell_type": "markdown",
   "metadata": {
    "nbsphinx": "hidden"
   },
   "source": [
    "# Random Signals\n",
    "\n",
    "*This jupyter notebook is part of a [collection of notebooks](../index.ipynb) on various topics of Digital Signal Processing. Please direct questions and suggestions to [Sascha.Spors@uni-rostock.de](mailto:Sascha.Spors@uni-rostock.de).*"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## White Noise"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Definition\n",
    "\n",
    "[White noise](https://en.wikipedia.org/wiki/White_noise) is a wide-sense stationary (WSS) random signal with constant power spectral density (PSD)\n",
    "\n",
    "\\begin{equation}\n",
    "\\Phi_{xx}(\\mathrm{e}^{\\,\\mathrm{j}\\, \\Omega}) = N_0\n",
    "\\end{equation}\n",
    "\n",
    "where $N_0$ denotes the power per frequency. White noise draws its name from the analogy to white light. It refers typically to an idealized model of a random signal, e.g. emerging from measurement noise. The auto-correlation function (ACF) of white noise can be derived by inverse discrete-time Fourier transformation (DTFT) of the PSD\n",
    "\n",
    "\\begin{equation}\n",
    "\\varphi_{xx}[\\kappa] = \\mathcal{F}_*^{-1} \\{ N_0 \\} = N_0 \\cdot \\delta[\\kappa]\n",
    "\\end{equation}\n",
    "\n",
    "This result implies that white noise has to be a zero-mean random process. It can be concluded from the ACF that two neighboring samples $k$ and $k+1$ are uncorrelated. Hence they show no dependencies in the statistical sense. Although this is often assumed, the probability density function (PDF) of white noise is not necessarily given by the normal distribution. In general, it is required to additionally state the amplitude distribution when denoting a signal as white noise."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Example - Amplifier Noise\n",
    "\n",
    "Additive white Gaussian noise (AWGN) is often used as a model for amplifier noise. In order to evaluate if this holds for a typical audio amplifier, the noise $n[k]$ captured from a microphone preamplifier at full amplification with open connectors is analyzed statistically. For the remainder, a function is defined to estimate and plot the PDF and ACF of a given random signal."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 1,
   "metadata": {},
   "outputs": [],
   "source": [
    "import numpy as np\n",
    "import matplotlib.pyplot as plt\n",
    "import scipy.stats as stats\n",
    "\n",
    "\n",
    "def estimate_plot_pdf_acf(x, nbins=50, acf_range=30):\n",
    "    \"\"\"Estimate and plot PDF/CDF of a given sample function.\"\"\"\n",
    "\n",
    "    # compute and truncate ACF\n",
    "    acf = 1 / len(x) * np.correlate(x, x, mode=\"full\")\n",
    "    acf = acf[len(x) - acf_range - 1 : len(x) + acf_range - 1]\n",
    "    kappa = np.arange(-acf_range, acf_range)\n",
    "\n",
    "    # plot PDF\n",
    "    plt.figure(figsize=(10, 6))\n",
    "    plt.subplot(121)\n",
    "    plt.hist(x, nbins, density=True)\n",
    "    plt.title(\"Estimated PDF\")\n",
    "    plt.xlabel(r\"$\\theta$\")\n",
    "    plt.ylabel(r\"$\\hat{p}_n(\\theta)$\")\n",
    "    plt.grid()\n",
    "\n",
    "    # plot ACF\n",
    "    plt.subplot(122)\n",
    "    plt.stem(kappa, acf)\n",
    "    plt.title(\"Estimated ACF\")\n",
    "    plt.ylabel(r\"$\\hat{\\varphi}_{nn}[\\kappa]$\")\n",
    "    plt.xlabel(r\"$\\kappa$\")\n",
    "    plt.axis([-acf_range, acf_range, 1.1 * min(acf), 1.1 * max(acf)])\n",
    "    plt.grid()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Now the pre-captured noise is loaded and analyzed"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 2,
   "metadata": {},
   "outputs": [
    {
     "data": {
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      "text/plain": [
       "<Figure size 960x576 with 2 Axes>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "noise = np.load(\"../data/amplifier_noise.npz\")[\"noise\"]\n",
    "estimate_plot_pdf_acf(noise, nbins=100, acf_range=150)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Inspecting the PDF reveals that it fits quite well to a [normal distribution](important_distributions.ipynb#Normal-Distribution). The ACF consists of a pronounced peak. from which can be concluded that the samples are approximately uncorrelated. Hence, the amplifier noise can be modeled reasonably well as additive white Gaussian noise. The parameters of the normal distribution (mean $\\mu_n$, variance $\\sigma_n^2$) are estimated as"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 3,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Mean:\t\t -1.537e-05 \n",
      "Variance:\t 1.140e-07\n"
     ]
    }
   ],
   "source": [
    "mean = np.mean(noise)\n",
    "variance = np.var(noise)\n",
    "\n",
    "print(\"Mean:\\t\\t {0:1.3e} \\nVariance:\\t {1:1.3e}\".format(mean, variance))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "**Excercise**\n",
    "\n",
    "* What relative level does the amplifier noise have when the maximum amplitude of the amplifier is assumed to be $\\pm 1$?\n",
    "\n",
    "Solution: The average power of a mean-free random signal is given by its variance, here $\\sigma_\\text{n}^2$. Due to the very low mean in comparison to the maximum amplitude, the noise can be assumed to be mean-free. Hence, the relative level of the noise is then given as $10*\\log_{10}\\left( \\frac{\\sigma_\\text{n}^2}{1} \\right)$. Numerical evaluation yields"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 4,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Level of amplifier noise: -69.43 dB\n"
     ]
    }
   ],
   "source": [
    "print(\"Level of amplifier noise: {:2.2f} dB\".format(10 * np.log10(variance / 1)))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Example - Generation of White Noise with Different Amplitude Distributions\n",
    "\n",
    "Toolboxes for numerical mathematics like `Numpy` or `scipy.stats` provide functions to draw uncorrelated random samples with a given amplitude distribution."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "**Uniformly distributed white noise**\n",
    "\n",
    "For samples drawn from a zero-mean random process with uniform amplitude distribution, the PDF and ACF are estimated as"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 5,
   "metadata": {},
   "outputs": [
    {
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      "text/plain": [
       "<Figure size 960x576 with 2 Axes>"
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     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "np.random.seed(3)\n",
    "estimate_plot_pdf_acf(np.random.uniform(size=10000) - 1 / 2)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Lets listen to uniformly distributed white noise"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 6,
   "metadata": {},
   "outputs": [],
   "source": [
    "from scipy.io import wavfile\n",
    "\n",
    "fs = 44100\n",
    "\n",
    "x = np.random.uniform(size=5 * fs) - 1 / 2\n",
    "wavfile.write(\"uniform_white_noise.wav\", fs, np.int16(x * 32768))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "<audio src=\"./uniform_white_noise.wav\" controls>Your browser does not support the audio element.</audio>[./uniform_white_noise.wav](./uniform_white_noise.wav)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "**Laplace distributed white noise**\n",
    "\n",
    "For samples drawn from a zero-mean random process with with Laplace amplitude distribution, the PDF and ACF are estimated as"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 7,
   "metadata": {},
   "outputs": [
    {
     "data": {
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nECgfg7WylOLRPBO8Dcmvld3zBrx1mdL5smp4HylCuMj/Rwdb1gp7ww/uE5zLa0XRUEwG6u643l7qUx6si9H6VacRnKNuRoGWn8daba+lkZoPEWe9FdRjQ8k+rF8Oio02WSgOyzoetQL4lfM5pdVW0AcKRmWML2t2ec6g/mCpi64X9iYknlUBs1srTJGcnztyFLZWy1IMjC4j7y3EK66ddZ73MxKutpiWuhRxPJ9WXW3EnrITCuKYE+UB64Z9QJ+tDdc0GDx1y8wnGlcLU4JBOQzlCS6vFgY+vcxn58GnK8Z+poAY2B1XNAIpf6RrTXP3uicSJopqzpbVoc+eoOtMc8sV15nPhRSa64NdpxuPa1H2lkrxsH6l+4r3NGTfhdjju55rsu+62j3dqKd7dvb7qwdUzzjd90SAnsiy74pZPcGwL8h2RO+OaaFnttn3zGM98+O+Z+btmdG7UoWeHKQjtenJmLpSsTNJ3p93k2fcKOctel8Ome7YQEpqA420iOOHMf8NNIqu3ShvrgHmt3/G48SnWz+fd4607wBKx2nbeel6pvis9ZUjxcEHE9BRCBsPkPUdKf7q4QDc12i6O9zNevCNOmP8dPc/ihKC1AplbmRzdHJlYW0KZW5kb2JqCjEyIDAgb2JqCjM4MTAKZW5kb2JqCjEwIDAgb2JqClsgXQplbmRvYmoKMTggMCBvYmoKPDwgL0JCb3ggWyAtMTAxNiAtMzUxIDE2NjAgMTA2OCBdIC9GaWx0ZXIgL0ZsYXRlRGVjb2RlIC9MZW5ndGggODEKL1N1YnR5cGUgL0Zvcm0gL1R5cGUgL1hPYmplY3QgPj4Kc3RyZWFtCnicNY27EcAwCEN7pmAEIyA2C+VS2Pu3MQ6p9E6nD4ld7NZ5ETQOTZJwVrRN1r08H1akCFaT9AR8Uu1TuDEGcmH3IJGkWpkj/92kh+4X3bkYGgplbmRzdHJlYW0KZW5kb2JqCjE5IDAgb2JqCjw8IC9GaWx0ZXIgL0ZsYXRlRGVjb2RlIC9MZW5ndGggMjE2ID4+CnN0cmVhbQp4nD1QMZIDMQjr/QqeYEBi7ffszU2Ku/+3kdlMClsMAiGohE1L6mOV3rQfH+RlmdP+B3aK+huYHywxcXW0LRGGy9VPYVqKvQcuWl5bmbJc68HNZsoQ6lEdkAZOvX2YPEowTNWvUP+D9zg+ThRHSRaD6pPre4QrM8Nc2sllvoTyeQ8Pttt8wOFGHNOR38glU945mdkaIFP0agn2gMxlpNbTQucycDTKbsWT2bv1GLpduBhSi2z2NSFTjd5MnTOovmRzqacgZTSjhWu2YkJKn/Pf4zV+3zsMTrMKZW5kc3RyZWFtCmVuZG9iagoyMCAwIG9iago8PCAvRmlsdGVyIC9GbGF0ZURlY29kZSAvTGVuZ3RoIDMzMyA+PgpzdHJlYW0KeJw1kkuOwzAMQ/c5hS4QwPr5c54OBrNI77+dJ7sFkkgxZYmk3T2lyW08bUjOJdmb/OgVq4v7kPfJZkrUayrRiTHkdUWExDAJjY36OFWvy2OeFQMdU2zx3wPEBrXZxBLEVcz5b7XHtDossdbE6a+QcVUQrQ701uHUrxObb8RFs2qJzLIWwiiaWUhVZUhnyPIK8II/SbXvg2kqcy+vlAUP7aLq8ApR5qAQllp6qdY10AI/r8mx0NCXbHeY+vXrdf2h5Kh9XxaopLmNJqnogXX6dohvJlTwrywPPZHOfe4s0Z5RL8h2KFG0UZwJPcflWQIqM/hl8Yvc+xVGIEGG3ESXMj0G0nNLc0FjGL7SBpJV7mB3KXG52eGcRUUMxbbKzKbcCGoY0/YJdHzgEDokmeQyjeU5zo16ru/dei7lumRUVof+yT5WPdj2+w9rBHnbCmVuZHN0cmVhbQplbmRvYmoKMjEgMCBvYmoKPDwgL0JCb3ggWyAtMTAxNiAtMzUxIDE2NjAgMTA2OCBdIC9GaWx0ZXIgL0ZsYXRlRGVjb2RlIC9MZW5ndGggMjg0Ci9TdWJ0eXBlIC9Gb3JtIC9UeXBlIC9YT2JqZWN0ID4+CnN0cmVhbQp4nD2RS24DIAwF95zCF4iEv8B5IlVdpPffdkzSrgBj+42fR1RIrJCfEWH3FrYkckrolpguz+HpUltew/e99M9OkkpUDwlxpmgcSXWxqZK0sNWl6Uc8iHiJHyejJHzeGhrlkdj6lv2APMf3iEWvVKDS8t6yOI3oAW4XWeXUxZRqhTpSZh/NmnDBlwsq5dQgqyVLBcKYJg9VcV/3fA47Rx42FyOazv9r7Pv/GrrfFereDXTOd0OglHf7gDPAvwGOQUTMmND7D2N3T2YwJo4ZZIkrbnWnw2JkY7WWbSaku7UfgY/MGNRRPUOcPrqbHek6H0GFoJm1mDjzZihL4ofJKDGEWFz3rA47VvZuWSkSycocYxvlz/pew9cvK29lNAplbmRzdHJlYW0KZW5kb2JqCjIyIDAgb2JqCjw8IC9CQm94IFsgLTEwMTYgLTM1MSAxNjYwIDEwNjggXSAvRmlsdGVyIC9GbGF0ZURlY29kZSAvTGVuZ3RoIDIzNQovU3VidHlwZSAvRm9ybSAvVHlwZSAvWE9iamVjdCA+PgpzdHJlYW0KeJw9ULttRDEM6z0FFwigv+15DARXXPZvQ/nlUomWKZLSiEx4TPwMjQe9h7rCYqOrLkH/qG+coeVYC7oLRMY+yxm2vYF79a/P+KOHFAUC4SSn42N2xmvEFPaFxjEX0gwxJ7LZRXVtVsRC1UIojXbAibue4fogo1czzPlmj1lkI6nUvqlMUnZ9zr/jm+6+DJNzdC+5qLmTOTPoXu3RKrmTmRQpgszJ3doj7z6F4N5aStXE7kM4R4w5ZeFLhankVh4u7aJ9/7ngQ8873uemWh/nys99/VT39VeKPomMA53RxW/mzxZ9z+9fQadWHQplbmRzdHJlYW0KZW5kb2JqCjE2IDAgb2JqCjw8IC9CYXNlRm9udCAvR0NXWERWK0RlamFWdVNhbnMtT2JsaXF1ZSAvQ2hhclByb2NzIDE3IDAgUgovRW5jb2RpbmcgPDwgL0RpZmZlcmVuY2VzIFsgMTEwIC9uIDExMiAvcCBdIC9UeXBlIC9FbmNvZGluZyA+PiAvRmlyc3RDaGFyIDAKL0ZvbnRCQm94IFsgLTEwMTYgLTM1MSAxNjYwIDEwNjggXSAvRm9udERlc2NyaXB0b3IgMTUgMCBSCi9Gb250TWF0cml4IFsgMC4wMDEgMCAwIDAuMDAxIDAgMCBdIC9MYXN0Q2hhciAyNTUKL05hbWUgL0dDV1hEVitEZWphVnVTYW5zLU9ibGlxdWUgL1N1YnR5cGUgL1R5cGUzIC9UeXBlIC9Gb250IC9XaWR0aHMgMTQgMCBSCj4+CmVuZG9iagoxNSAwIG9iago8PCAvQXNjZW50IDkyOSAvQ2FwSGVpZ2h0IDAgL0Rlc2NlbnQgLTIzNiAvRmxhZ3MgOTYKL0ZvbnRCQm94IFsgLTEwMTYgLTM1MSAxNjYwIDEwNjggXSAvRm9udE5hbWUgL0dDV1hEVitEZWphVnVTYW5zLU9ibGlxdWUKL0l0YWxpY0FuZ2xlIDAgL01heFdpZHRoIDEzNTAgL1N0ZW1WIDAgL1R5cGUgL0ZvbnREZXNjcmlwdG9yIC9YSGVpZ2h0IDAgPj4KZW5kb2JqCjE0IDAgb2JqClsgNjAwIDYwMCA2MDAgNjAwIDYwMCA2MDAgNjAwIDYwMCA2MDAgNjAwIDYwMCA2MDAgNjAwIDYwMCA2MDAgNjAwIDYwMCA2MDAKNjAwIDYwMCA2MDAgNjAwIDYwMCA2MDAgNjAwIDYwMCA2MDAgNjAwIDYwMCA2MDAgNjAwIDYwMCAzMTggNDAxIDQ2MCA4MzggNjM2Cjk1MCA3ODAgMjc1IDM5MCAzOTAgNTAwIDgzOCAzMTggMzYxIDMxOCAzMzcgNjM2IDYzNiA2MzYgNjM2IDYzNiA2MzYgNjM2IDYzNgo2MzYgNjM2IDMzNyAzMzcgODM4IDgzOCA4MzggNTMxIDEwMDAgNjg0IDY4NiA2OTggNzcwIDYzMiA1NzUgNzc1IDc1MiAyOTUKMjk1IDY1NiA1NTcgODYzIDc0OCA3ODcgNjAzIDc4NyA2OTUgNjM1IDYxMSA3MzIgNjg0IDk4OSA2ODUgNjExIDY4NSAzOTAgMzM3CjM5MCA4MzggNTAwIDUwMCA2MTMgNjM1IDU1MCA2MzUgNjE1IDM1MiA2MzUgNjM0IDI3OCAyNzggNTc5IDI3OCA5NzQgNjM0IDYxMgo2MzUgNjM1IDQxMSA1MjEgMzkyIDYzNCA1OTIgODE4IDU5MiA1OTIgNTI1IDYzNiAzMzcgNjM2IDgzOCA2MDAgNjM2IDYwMCAzMTgKMzUyIDUxOCAxMDAwIDUwMCA1MDAgNTAwIDEzNTAgNjM1IDQwMCAxMDcwIDYwMCA2ODUgNjAwIDYwMCAzMTggMzE4IDUxOCA1MTgKNTkwIDUwMCAxMDAwIDUwMCAxMDAwIDUyMSA0MDAgMTAyOCA2MDAgNTI1IDYxMSAzMTggNDAxIDYzNiA2MzYgNjM2IDYzNiAzMzcKNTAwIDUwMCAxMDAwIDQ3MSA2MTcgODM4IDM2MSAxMDAwIDUwMCA1MDAgODM4IDQwMSA0MDEgNTAwIDYzNiA2MzYgMzE4IDUwMAo0MDEgNDcxIDYxNyA5NjkgOTY5IDk2OSA1MzEgNjg0IDY4NCA2ODQgNjg0IDY4NCA2ODQgOTc0IDY5OCA2MzIgNjMyIDYzMiA2MzIKMjk1IDI5NSAyOTUgMjk1IDc3NSA3NDggNzg3IDc4NyA3ODcgNzg3IDc4NyA4MzggNzg3IDczMiA3MzIgNzMyIDczMiA2MTEgNjA4CjYzMCA2MTMgNjEzIDYxMyA2MTMgNjEzIDYxMyA5OTUgNTUwIDYxNSA2MTUgNjE1IDYxNSAyNzggMjc4IDI3OCAyNzggNjEyIDYzNAo2MTIgNjEyIDYxMiA2MTIgNjEyIDgzOCA2MTIgNjM0IDYzNCA2MzQgNjM0IDU5MiA2MzUgNTkyIF0KZW5kb2JqCjE3IDAgb2JqCjw8IC9uIDE5IDAgUiAvcCAyMCAwIFIgPj4KZW5kb2JqCjI3IDAgb2JqCjw8IC9GaWx0ZXIgL0ZsYXRlRGVjb2RlIC9MZW5ndGggOTEgPj4Kc3RyZWFtCnicNYy7DcAwCER7prgR+DiA94miFPb+bYgtF9w96YnzbGBknYcjtOMWsqZwU0xSTqh3DGqlNx076CXN/TTJei4a9A9x9RW2mwOSUSSRh0SXy5Vn5V98PgxvHGIKZW5kc3RyZWFtCmVuZG9iagoyOCAwIG9iago8PCAvRmlsdGVyIC9GbGF0ZURlY29kZSAvTGVuZ3RoIDIzNSA+PgpzdHJlYW0KeJw1UUluADEIu+cV/kClsCfvmarqof3/tYZRLwMD2Ngk78FGJD7EkO4oV3zK6jTL8DtZ5MXPSuHkvYgKpCrCCmkHz3JWMwyeG5kClzPxWWY+mRY7FlBNxHF25DSDQYhpXEfL6TDTPOgJuT4YcWOnWa5iSOvdUr2+1/KfKspH1t0st07Z1ErdomfsSVx2Xk9taV8YdRQ3BZEOHzu8B/ki5iwuOpFu9psph5WkITgtgB+JoVTPDq8RJn5mJHjKnk7vozS89kHT9b17QUduJmQqt1BGKp6sNMaMofqNaCap7/+BnvW9vv4AQ01UuQplbmRzdHJlYW0KZW5kb2JqCjI5IDAgb2JqCjw8IC9GaWx0ZXIgL0ZsYXRlRGVjb2RlIC9MZW5ndGggMTY0ID4+CnN0cmVhbQp4nD2QwRFDIQhE71axJYCAQD3JZHL4v/9rQJNcZB1g96k7gZBRhzPDZ+LJg9OxNHBvFYxrCK8j9AhNApPAxMGaeAwLAadhkWMu31WWVaeVrpqNnte9Y0HVaZc1DW3agfKtjz/CNd6j8BrsHkIHsSh0bmVaC5lYPGucO8yjzOd+Ttt3PRitptSsN3LZ1z06y9RQXlr7hM5otP0n1y+7MV4fhRQ5CAplbmRzdHJlYW0KZW5kb2JqCjMwIDAgb2JqCjw8IC9GaWx0ZXIgL0ZsYXRlRGVjb2RlIC9MZW5ndGggODEgPj4Kc3RyZWFtCnicTc27DcAgDATQnik8AuD/PlGUItm/jQ0RobGfdCedYIcKbnFYDLQ7HK341FOYfegeEpJQc91EWDMl2oSkX/rLMMOYWMi2rzdXrnK+FtwciwplbmRzdHJlYW0KZW5kb2JqCjMxIDAgb2JqCjw8IC9GaWx0ZXIgL0ZsYXRlRGVjb2RlIC9MZW5ndGggNzYgPj4Kc3RyZWFtCnicMzU3VTBQsLQAEqaG5grmRpYKKYZcQD6IlcsFE8sBs8xMzIAsQ0tklomxIZBlYmGGxDI2sYDKIlgGQBpsTQ7M9ByuDK40ADUXGQUKZW5kc3RyZWFtCmVuZG9iagozMiAwIG9iago8PCAvRmlsdGVyIC9GbGF0ZURlY29kZSAvTGVuZ3RoIDE3MCA+PgpzdHJlYW0KeJw9kEsSwyAMQ/ecQkcA/4DztNPpgtx/W8uZdIMUY8svRFd07JWHx8aUjfdoY0+ELVzldBpOUxmPi7tmXaDLYTLTb7yaucBUYZHV7KL6GLyh86xmh69VMzGEN5kSGmAqd3IP9fWnOO3bkpBsV2HQnRqkszDMkfw9EFNz0HOIkfwjX3JrYdCZ5hcXLasZrWVM0exhqmwtDOqNQXfK9dR6rvMwEe/zA99BPmQKZW5kc3RyZWFtCmVuZG9iagozMyAwIG9iago8PCAvRmlsdGVyIC9GbGF0ZURlY29kZSAvTGVuZ3RoIDMwNyA+PgpzdHJlYW0KeJw9kktuAzEMQ/c+hS4QwPrZnvOkKLqY3n/bJyXpihzZFkVqlrpMWVMekDSThH/p8HCxnfI7bM9mZuBaopeJ5ZTn0BVi7qJ82cxGXVknxeqEZjq36FE5Fwc2Taqfqyyl3S54Dtcmnlv2ET+80KAe1DUuCTd0V6NlKTRjqvt/0nv8jDLgakxdbFKrex88XkRV6OgHR4kiY5cX5+NBCelKwmhaiJV3RQNB7vK0ynsJ7tveasiyB6mYzjspZrDrdFIubheHIR7I8qjw5aPYa0LP+LArJfRI2IYzcifuaMbm1MjikP7ejQRLj65oIfPgr27WLmC8UzpFYmROcqxpi1VO91AU07nDvQwQ9WxFQylzkdXqX8POC2uWbBZ4SvoFHqPdJksOVtnbqE7vrTzZ0PcfWtd0HwplbmRzdHJlYW0KZW5kb2JqCjM0IDAgb2JqCjw8IC9GaWx0ZXIgL0ZsYXRlRGVjb2RlIC9MZW5ndGggNzMgPj4Kc3RyZWFtCnicM7Y0UDBQsDBT0DU0NlQwsjRWMDczUEgx5AIKgVi5XDCxHDDLzBLEMjQ3Q2LpmhlCZZFYIONyuGAG58DMy+HK4EoDAB6JFpUKZW5kc3RyZWFtCmVuZG9iagozNSAwIG9iago8PCAvRmlsdGVyIC9GbGF0ZURlY29kZSAvTGVuZ3RoIDY5ID4+CnN0cmVhbQp4nDO2NFAwULA0V9A1NDZUMDYwUTA3M1BIMeSCMXPBLLBsDhdMHYRlBmIYGZogscyAxoEl4QyQGTlw03K4MrjSAPqpFkUKZW5kc3RyZWFtCmVuZG9iagozNiAwIG9iago8PCAvRmlsdGVyIC9GbGF0ZURlY29kZSAvTGVuZ3RoIDIzMSA+PgpzdHJlYW0KeJw1TzmSBCEMy3mFPjBVGNtAv6entjbY+X+6kplOkPAhydMTHZl4mSMjsGbH21pkIGbgU0zFv/a0DxOq9+AeIpSLC2GGkXDWrONuno4X/3aVz1gH7zb4illeENjCTNZXFmcu2wVjaZzEOclujF0TsY11radTWEcwoQyEdLbDlCBzVKT0yY4y5ug4kSeei+/22yx2OX4O6ws2jSEV5/gqeoI2g6Lsee8CGnJB/13d+B5Fu+glIBsJFtZRYu6c5YRfvXZ0HrUoEnNCmkEuEyHN6SqmEJpQrLOjoFJRcKk+p+isn3/lX1wtCmVuZHN0cmVhbQplbmRvYmoKMzcgMCBvYmoKPDwgL0ZpbHRlciAvRmxhdGVEZWNvZGUgL0xlbmd0aCAyNDkgPj4Kc3RyZWFtCnicPVA7jkQhDOs5hS/wJPIjcB5Gqy1m79+uA5opUEx+tjMk0BGBRwwxlK/jJa2groG/i0LxbuLrg8Igq0NSIM56D4h07KY2kRM6HZwzP2E3Y47ARTEGnOl0pj0HJjn7wgqEcxtl7FZIJ4mqIo7qM44pnip7n3gWLO3INlsnkj3kIOFSUonJpZ+Uyj9typQKOmbRBCwSueBkE004y7tJUowZlDLqHqZ2In2sPMijOuhkTc6sI5nZ00/bmfgccLdf2mROlcd0Hsz4nLTOgzkVuvfjiTYHTY3a6Oz3E2kqL1K7HVqdfnUSld0Y5xgSl2d/Gd9k//kH/odaIgplbmRzdHJlYW0KZW5kb2JqCjM4IDAgb2JqCjw8IC9GaWx0ZXIgL0ZsYXRlRGVjb2RlIC9MZW5ndGggMzk1ID4+CnN0cmVhbQp4nD1SS27FQAjb5xRcoNLwm895UlXdvPtva0NSqSq8iTHGMH3KkLnlS10ScYXJt16uWzymfC5bWpl5iLuLjSU+ttyX7iG2XXQusTgdR/ILMp0qRKjNqtGh+EKWhQeQTvChC8J9Of7jL4DB17ANuOE9MkGwJOYpQsZuURmaEkERYeeRFaikUJ9Zwt9R7uv3MgVqb4ylC2Mc9Am0BUJtSMQC6kAAROyUVK2QjmckE78V3WdiHGDn0bIBrhlURJZ77MeIqc6ojLxExD5PTfoolkwtVsZuUxlf/JSM1Hx0BSqpNPKU8tBVs9ALWIl5EvY5/Ej459ZsIYY6btbyieUfM8UyEs5gSzlgoZfjR+DbWXURrh25uM50gR+V1nBMtOt+yPVP/nTbWs11vHIIokDlTUHwuw6uRrHExDI+nY0peqIssBqavEYzwWEQEdb3w8gDGv1yvBA0p2sitFgim7ViRI2KbHM9vQTWTO/FOdbDE8Js753WobIzMyohgtq6hmrrQHazvvNwtp8/M+iibQplbmRzdHJlYW0KZW5kb2JqCjM5IDAgb2JqCjw8IC9GaWx0ZXIgL0ZsYXRlRGVjb2RlIC9MZW5ndGggMjQ5ID4+CnN0cmVhbQp4nE1RSYoDMAy75xX6QCFek7ynQ5lD5//Xyg6FOQQJr5KTlphYCw8xhB8sPfiRIXM3/Rt+otm7WXqSydn/mOciU1H4UqguYkJdiBvPoRHwPaFrElmxvfE5LKOZc74HH4W4BDOhAWN9STK5qOaVIRNODHUcDlqkwrhrYsPiWtE8jdxu+0ZmZSaEDY9kQtwYgIgg6wKyGCyUNjYTMlnOA+0NyQ1aYNepG1GLgiuU1gl0olbEqszgs+bWdjdDLfLgqH3x+mhWl2CF0Uv1WHhfhT6YqZl27pJCeuFNOyLMHgqkMjstK7V7xOpugfo/y1Lw/cn3+B2vD838XJwKZW5kc3RyZWFtCmVuZG9iago0MCAwIG9iago8PCAvRmlsdGVyIC9GbGF0ZURlY29kZSAvTGVuZ3RoIDk0ID4+CnN0cmVhbQp4nEWNwRHAIAgE/1RBCQoK2k8mk4f2/40QMnxg5w7uhAULtnlGHwWVJl4VWAdKY9xQj0C94XItydwFD3Anf9rQVJyW03dpkUlVKdykEnn/DmcmkKh50WOd9wtj+yM8CmVuZHN0cmVhbQplbmRvYmoKNDEgMCBvYmoKPDwgL0ZpbHRlciAvRmxhdGVEZWNvZGUgL0xlbmd0aCA3MiA+PgpzdHJlYW0KeJwzMrdQMFCwNAEShhYmCuZmBgophlxAvqmJuUIuF0gMxMoBswyAtCWcgohngJggbRDFIBZEsZmJGUQdnAGRy+BKAwAl2xbJCmVuZHN0cmVhbQplbmRvYmoKNDIgMCBvYmoKPDwgL0ZpbHRlciAvRmxhdGVEZWNvZGUgL0xlbmd0aCAyNTggPj4Kc3RyZWFtCnicRZFLcgQgCET3noIjgPzkPJNKZTG5/zYNzmQ2dpeo/YRKI6YSLOcUeTB9yfLNZLbpdzlWOxsFFEUomMlV6LECqztTxJlriWrrY2XkuNM7BsUbzl05qWRxo4x1VHUqcEzPlfVR3fl2WZR9Rw5lCtiscxxs4MptwxgnRput7g73iSBPJ1NHxe0g2fAHJ419lasrcJ1s9tFLMA4E/UITmOSLQOsMgcbNU/TkEuzj43bngWBveRFI2RDIkSEYHYJ2nVz/4tb5vf9xhjvPtRmuHO/id5jWdsdfYpIVcwGL3Cmo52suWtcZOt6TM8fkpvuGzrlgl7uDTO/5P9bP+v4DHilm+gplbmRzdHJlYW0KZW5kb2JqCjQzIDAgb2JqCjw8IC9CQm94IFsgLTEwMjEgLTQ2MyAxNzk0IDEyMzMgXSAvRmlsdGVyIC9GbGF0ZURlY29kZSAvTGVuZ3RoIDM5Ci9TdWJ0eXBlIC9Gb3JtIC9UeXBlIC9YT2JqZWN0ID4+CnN0cmVhbQp4nOMyNDBTMDY1VcjlMjc2ArNywCwjcyMgCySLYEFkM7jSABXzCnwKZW5kc3RyZWFtCmVuZG9iago0NCAwIG9iago8PCAvRmlsdGVyIC9GbGF0ZURlY29kZSAvTGVuZ3RoIDgzID4+CnN0cmVhbQp4nEWMuw3AMAhEe6ZgBH4m9j5RlMLevw0QJW64J909XB0JmSluM8NDBp4MLIZdcYH0ljALXEdQjp3so2HVvuoEjfWmUvPvD5Se7KzihusBAkIaZgplbmRzdHJlYW0KZW5kb2JqCjQ1IDAgb2JqCjw8IC9GaWx0ZXIgL0ZsYXRlRGVjb2RlIC9MZW5ndGggMTUwID4+CnN0cmVhbQp4nD1POQ7DMAzb/Qp+IIB1WLbekyLokP5/reigHQQRoHjIsqNjBQ4xgUnHHImXtB/8NI1ALIOaI3pCxTAscDZJgYdBVklXPlucTCG1BV5I+NYUPBtNZIKex4gduJPJ6APvUluxdegB7RBVJF2zDGOxrbpwPWEENcV6Feb1MGOJkkefVUvrAQUT+NL5f+5u73Z9AWDFMF0KZW5kc3RyZWFtCmVuZG9iago0NiAwIG9iago8PCAvRmlsdGVyIC9GbGF0ZURlY29kZSAvTGVuZ3RoIDE1MSA+PgpzdHJlYW0KeJw1j8sNwzAMQ++aggsE0M+yPE+KoId0/2slpwUMmDDJJ9kWg5GMQ0xg7Jhj4SVUT60+JCO3ukk5EKlQNwRPaEwMM5ykS+CV6b5lPLd4Oa3UBZ2QyN1ZbTREGc08hqOn7BVO+i9zk6iVGWUOyKxT3U7IyoIVmBfUct/PuK1yoRMetivDojvFiHKbGWn4/e+kN11f3Hgw1QplbmRzdHJlYW0KZW5kb2JqCjQ3IDAgb2JqCjw8IC9GaWx0ZXIgL0ZsYXRlRGVjb2RlIC9MZW5ndGggNTEgPj4Kc3RyZWFtCnicMza0UDBQMDQwB5JGhkCWkYlCiiEXSADEzOWCCeaAWQZAGqI4B64mhyuDKw0A4bQNmAplbmRzdHJlYW0KZW5kb2JqCjQ4IDAgb2JqCjw8IC9GaWx0ZXIgL0ZsYXRlRGVjb2RlIC9MZW5ndGggMzM0ID4+CnN0cmVhbQp4nC1SS3LFIAzbcwpdoDP4B+Q86XS6eL3/tpKTRUYOYPQx5YaJSnxZILej1sS3jcxAheGvq8yFz0jbyDqIy5CLuJIthXtELOQxxDzEgu+r8R4e+azMybMHxi/Zdw8r9tSEZSHjxRnaYRXHYRXkWLB1Iap7eFOkw6kk2OOL/z7Fcy0ELXxG0IBf5J+vjuD5khZp95ht0656sEw7qqSwHGxPc14mX1pnuToezwfJ9q7YEVK7AhSFuTPOc+Eo01ZGtBZ2NkhqXGxvjv1YStCFblxGiiOQn6kiPKCkycwmCuKPnB5yKgNh6pqudHIbVXGnnsw1m4u3M0lm675IsZnCeV04s/4MU2a1eSfPcqLUqQjvsWdL0NA5rp69lllodJsTvKSEz8ZOT06+VzPrITkVCaliWlfBaRSZYgnbEl9TUVOaehn++/Lu8Tt+/gEsc3xzCmVuZHN0cmVhbQplbmRvYmoKNDkgMCBvYmoKPDwgL0ZpbHRlciAvRmxhdGVEZWNvZGUgL0xlbmd0aCAzMjAgPj4Kc3RyZWFtCnicNVJLbgUxCNvPKbhApfBPzvOqqou++29rE70VTDBg4ykvWdJLvtQl26XD5Fsf9yWxQt6P7ZrMUsX3FrMUzy2vR88Rty0KBFETPViZLxUi1M/06DqocEqfgVcItxQbvINJAINq+AcepTMgUOdAxrtiMlIDgiTYc2lxCIlyJol/pLye3yetpKH0PVmZy9+TS6XQHU1O6AHFysVJoF1J+aCZmEpEkpfrfbFC9IbAkjw+RzHJgOw2iW2iBSbnHqUlzMQUOrDHArxmmtVV6GDCHocpjFcLs6gebPJbE5WkHa3jGdkw3sswU2Kh4bAF1OZiZYLu5eM1r8KI7VGTXcNw7pbNdwjRaP4bFsrgYxWSgEensRINaTjAiMCeXjjFXvMTOQ7AiGOdmiwMY2gmp3qOicDQnrOlYcbHHlr18w9U6XyHCmVuZHN0cmVhbQplbmRvYmoKNTAgMCBvYmoKPDwgL0ZpbHRlciAvRmxhdGVEZWNvZGUgL0xlbmd0aCAxOCA+PgpzdHJlYW0KeJwzNrRQMIDDFEOuNAAd5gNSCmVuZHN0cmVhbQplbmRvYmoKNTEgMCBvYmoKPDwgL0ZpbHRlciAvRmxhdGVEZWNvZGUgL0xlbmd0aCAxMzMgPj4Kc3RyZWFtCnicRY9LDgQhCET3nKKOwMcf53Ey6YVz/+2AnW4TYz2FVIG5gqE9LmsDnRUfIRm28beplo5FWT5UelJWD8ngh6zGyyHcoCzwgkkqhiFQi5gakS1lbreA2zYNsrKVU6WOsIujMI/2tGwVHl+iWyJ1kj+DxCov3OO6Hcil1rveoou+f6QBMQkKZW5kc3RyZWFtCmVuZG9iago1MiAwIG9iago8PCAvRmlsdGVyIC9GbGF0ZURlY29kZSAvTGVuZ3RoIDM0MCA+PgpzdHJlYW0KeJw1UjluBDEM6/0KfSCAbtvv2SBIkfy/DanZFANxdFKUO1pUdsuHhVS17HT5tJXaEjfkd2WFxAnJqxLtUoZIqLxWIdXvmTKvtzVnBMhSpcLkpORxyYI/w6WnC8f5trGv5cgdjx5YFSOhRMAyxcToGpbO7rBmW36WacCPeIScK9Ytx1gFUhvdOO2K96F5LbIGiL2ZlooKHVaJFn5B8aBHjX32GFRYINHtHElwjIlQkYB2gdpIDDl7LHZRH/QzKDET6NobRdxBgSWSmDnFunT03/jQsaD+2Iw3vzoq6VtaWWPSPhvtlMYsMul6WPR089bHgws076L859UMEjRljZLGB63aOYaimVFWeLdDkw3NMcch8w6ewxkJSvo8FL+PJRMdlMjfDg2hf18eo4ycNt4C5qI/bRUHDuKzw165gRVKF2uS9wGpTOiB6f+v8bW+19cfHe2AxgplbmRzdHJlYW0KZW5kb2JqCjUzIDAgb2JqCjw8IC9GaWx0ZXIgL0ZsYXRlRGVjb2RlIC9MZW5ndGggMjUxID4+CnN0cmVhbQp4nC1RSXIDQQi7zyv0hGan32OXK4fk/9cIygcGDYtAdFrioIyfICxXvOWRq2jD3zMxgt8Fh34r121Y5EBUIEljUDWhdvF69B7YcZgJzJPWsAxmrA/8jCnc6MXhMRlnt9dl1BDsXa89mUHJrFzEJRMXTNVhI2cOP5kyLrRzPTcg50ZYl2GQblYaMxKONIVIIYWqm6TOBEESjK5GjTZyFPulL490hlWNqDHscy1tX89NOGvQ7Fis8uSUHl1xLicXL6wc9PU2AxdRaazyQEjA/W4P9XOyk994S+fOFtPje83J8sJUYMWb125ANtXi37yI4/uMr+fn+fwDX2BbiAplbmRzdHJlYW0KZW5kb2JqCjU0IDAgb2JqCjw8IC9CQm94IFsgLTEwMjEgLTQ2MyAxNzk0IDEyMzMgXSAvRmlsdGVyIC9GbGF0ZURlY29kZSAvTGVuZ3RoIDc0Ci9TdWJ0eXBlIC9Gb3JtIC9UeXBlIC9YT2JqZWN0ID4+CnN0cmVhbQp4nOPSNbIwVbAwMFDI5dI1MjQGM3O4dC2NFcwMzUAsQzNDGNPIxFLB3BjMNDY2h4maGJjCFUDNyuHKgKg2NYMYDGWCxNMA6/UWMgplbmRzdHJlYW0KZW5kb2JqCjU1IDAgb2JqCjw8IC9GaWx0ZXIgL0ZsYXRlRGVjb2RlIC9MZW5ndGggMjE1ID4+CnN0cmVhbQp4nDVROQ4DIQzs9xX+QCSML3hPoijN/r/NjNFWHsFchrSUIZnyUpOoIeVTPnqZLpy63NfMajTnlrQtc4C4trwvrZLAiWaIg8FpmLgBmjwBQ9fRqFFDFx7Q1KVTKLDcBD6Kt24P3WO1gZe2IeeJIGIoGSxBzalFExZtzyekNb9eixvel+3dyFOlxpYYgQYBVjgc1+jX8JU9TybRdBUy1Ks1yxgJE0UiPPmOptUT61o00jIS1MYRrGoDvDv9ME4AABNxywJkn0qUs+TEb7H0swZX+v4Bn0dUlgplbmRzdHJlYW0KZW5kb2JqCjI1IDAgb2JqCjw8IC9CYXNlRm9udCAvQk1RUURWK0RlamFWdVNhbnMgL0NoYXJQcm9jcyAyNiAwIFIKL0VuY29kaW5nIDw8Ci9EaWZmZXJlbmNlcyBbIDMyIC9zcGFjZSA0MCAvcGFyZW5sZWZ0IC9wYXJlbnJpZ2h0IDQ2IC9wZXJpb2QgNDggL3plcm8gL29uZSAvdHdvIC90aHJlZQovZm91ciAvZml2ZSAvc2l4IDU2IC9laWdodCA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   "source": [
    "estimate_plot_pdf_acf(np.random.laplace(size=10000, loc=0, scale=1 / np.sqrt(2)))"
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   "cell_type": "markdown",
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   "source": [
    "**Exercise**\n",
    "\n",
    "* Do both random processes represent white noise?\n",
    "* Estimate the power spectral density $N_0$ of both examples.\n",
    "* How does the ACF change if you lower the length `size` of the random signal. Why?\n",
    "\n",
    "Solution: Both processes represent white noise since the ACF can be approximated reasonably well as Dirac impulse $\\delta[\\kappa]$. The weight of the Dirac impulse is equal to $N_0$. In case of the uniformly distributed white noise $N_0 \\approx \\frac{1}{12}$, in case of the Laplace distributed white noise $N_0 \\approx 1$. Decreasing the length `size` of the signal increases the statistical uncertainties in the estimate of the ACF.  "
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   "source": [
    "**Copyright**\n",
    "\n",
    "This notebook is provided as [Open Educational Resource](https://en.wikipedia.org/wiki/Open_educational_resources). Feel free to use the notebook for your own purposes. The text is licensed under [Creative Commons Attribution 4.0](https://creativecommons.org/licenses/by/4.0/), the code of the IPython examples under the [MIT license](https://opensource.org/licenses/MIT). Please attribute the work as follows: *Sascha Spors, Digital Signal Processing - Lecture notes featuring computational examples*."
   ]
  }
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